Biomedical Engineering Reference
In-Depth Information
difficulties linked to the moving domain. In particular, one can refer to [
43
]for
the ALE method, [
85
] for the fictitious domain method, or [
133
] for the immersed
boundary method.
We will focus here on the fluid-structure coupling. Indeed, even if we consider
a linear coupled problem, one major issue is how to efficiently discretize in
time the coupling conditions (
1.15
)and(
1.17
)? The
implicit
(or
strongly coupled
schemes
) are stable since they preserve the energy balance at the interface. The
explicit
(or staggered) scheme are cheaper but do not preserve the energy balance
at the interface and may lead to numerical instabilities in particular in the case
of strong added mass effect of the fluid on the structure (i.e.,
f
close to
s
),
cf. also Proposition
1.1
. In order to balance out the energy at the interface and
to stabilize explicitly coupled schemes typically inner sub-iterations are needed,
see, e.g., [
25
,
37
,
63
,
69
,
72
,
130
,
131
,
140
], and the references therein. Semi-implicit
schemes have been introduced based on the implicit treatment of the added mass
effect and the explicit treatment of the viscous stress [
63
]. In the case of a thin
structure there is however also another strategy to solve fluid-structure interaction
problem without inner sub-iterations. The so-called kinematically coupled schemes
treat implicitly only the hydrodynamic fluid-structure coupling (i.e., the added mass
effect), whereas the contribution of the elastic structure is treated explicitly. We
refer, e.g., to [
23
,
56
,
57
,
62
,
98
,
110
,
121
] and to our more detailed discussion in
Sect.
1.3
. See also [
82
].
The goal of this chapter is to enlighten some of these difficulties, some strategies
to overcome them and to present few open questions. Consequently we will review
some of the existence results that can be found, as well as some of the numerical
schemes. We will only give the key ideas and steps of these results and the reader
may refer to corresponding papers for the details.
1.2
Mathematical Analysis
Last years, existence of weak or strong solutions for fluid-structure coupled
problems have been the object of numerous researches.
A vast majority of works concern a rigid solid moving in a viscous incompress-
ible Newtonian fluid whose behavior is described by the equations of Navier-Stokes
(historically, the weak formulation of the problem of the motion of rigid bodies in
viscous fluids has been introduced and studied in [
112
], and further in [
33
,
39
,
40
,
75
,
89
,
107
,
108
,
146
,
147
,
149
,
153
,
154
] for existence of weak or strong solutions).
Note that, in these cases, the displacement of the structure remains regular enough
and that we have a parabolic-ODE coupling. We refer to Chap.
2
for further details.
In these problems a challenging point is the existence of collisions. Let us first
mention that in the case of compressible fluids this problem has been clarified in [
52
,
Lemma 3.1, Corollary 3.1] where it is proven that solids are allowed to touch but not
to penetrate one another unless they did so at the initial time. In the incompressible
case the situation is different.
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